Rainbow connectivity of multilayered random geometric graphs

An edge-colored multigraph G is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from h≥ 2 independent random geometric graphs $G(n,r)$ on the unit square. We define an edge-coloring by coloring the edges according to the copy of $G(n,r)$ they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that r(n)=(log n/n)h-1/2h is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erd{o}s-R\' enyi random model.